On the polar derivative of lacunary type polynomials

Abstract

Let $p(z)=a_nz^n+\sum_{l=\nu}^na_{n-l}z^{n-l}$, where $1\leq \nu \leq n$, be a polynomial of degree $n$ having all its zeros in $|z|\leq k\leq 1$. For polar derivative $D_{\alpha}p(z)$, it is known that for each $|\alpha|\leq 1$ on $|z|=1$, \begin{align*} |D_{\alpha}p(z)|\leq \frac{n}{1+k^{\nu}}\Big\{(|\alpha|+k^{\nu})\|p\|_{\infty}-\frac{1-|\alpha|}{k^{n-\nu}}\min_{|z|=k}|p(z)|\Big\}. \end{align*} In this paper, we obtain the $L_q$ mean extension and a refinement of the above and other related results for the polar derivative of polynomials.

Keywords

• Derivative
• lacunary type polynomial
• $L^q$ Inequality
• maximum modulus
• restricted zeros

References

1. Bernstein, S., Sur la limitation des derivees des polnomese, C. R. Acad. Sci. Paris., 190, (1930), 338-341.
2. Zygmund, A., A remark on conjugate series, Proc. London Math. Soc., 34 (1932), 392-400. https://doi.org/10.1112/plms/s2-34.1.392
3. Arestov, V. V. , On integral inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat., 45 (1981), 3-22 (in Russian), English transl. in Math. USSR Izv., 18 (1982), 1-17. https://doi.org/10.1070/IM1982v018n01ABEH001375
4. Lax, P. D., Proof of a conjecture of P. Erd¨os on the derivative of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513. https://doi.org/10.1090/S0002-9904-1944-08177-9
5. Turan, P., Über die ableitung von Polynomen, Compos. Math., 7 (1939), 89-95.
6. De-Bruijn, N. G., Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch. Proc., 50, (1947), 1265-1272.
7. Rahman, Q. I., Schmeisser, G., Lp inequalities for polynomials, J. Approx. Theory., 53 (1988), 26-32. https://doi.org/10.1016/0021-9045(88)90073-1
8. Malik, M. A., On the derivative of a polynomial, J. London Math. Soc., 1 (1969), 57-60. http://doi.org/10.1112/jlms/s2-1.1.57

9. Govil, N. K., Rahman, Q. I., Functions of exponential type not vanishing in a half-plane and related polynomials, Trans. Amer. Math. Soc., 137 (1969), 501-517. https://doi.org/10.1090/S0002-9947-1969-0236385-6
10. Qazi, M. A., On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115 (1992), 337-343. https://doi.org/10.1090/S0002-9939-1992-1113648-1
11. Gardner, R. B., Weems, A., A Bernstein type Lp inequality for a certain class of polynomials, J. Math. Anal. Appl., 219 (1998), 472-478. https://doi.org/10.1006/jmaa.1997.5838
12. Aziz, A., Shah, W. M., An integral mean estimate for polynomial, Indian J. Pure Appl. Math., 28 (1997), 1413-1419.
13. Aziz, A., Rather, N. A., On an inequality concerning the polar derivative of a polynomial, Proc. Math. Sci., 117, (2007), 349-357. https://doi.org/10.48550/arXiv.0709.3346
14. Rather, N. A., Iqbal, A., Hyun, G. H., Integral inequalities for the polar derivative of a polynomial, Nonlinear Funct. Anal. Appl., 23 (2018), 381-393.
15. Dewan, K. K., Singh, N., Mir, A., Extensions of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl., 352 (2009), 807-815. https://doi.org/10.1016/j.jmaa.2008.10.056
16. Gardner, R. B., Govil, N. K., Weems, A., Some results concerning rate of growth of polynomials, East J. Approx., 10 (2004), 301-312.
17. Aziz, A., Rather, N. A., Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl., 289 (2004), 14-29. https://doi.org/10.1016/S0022-247X(03)00530-4
18. Zireh, A., On the polar derivative of a polynomial, Bull. Iranian. Math. Soc., 41(2014), 967-976.
19. Aziz, A., Rather, N. A., New Lp inequalities for polynomials, J. Math. Inequl. App., 1 (1998), 177-191.